Semi-analytical solutions for eigenvalue problems of chains and periodic graphs

Abstract

We show the existence and nature of convergence to a limiting set of roots for polynomials in a three-term recurrence of the form pn+1(z)=Qk(z)pn(z)+γpn−1(z) as n→∞, where the coefficient Qk(z) is a kth degree polynomial, and z,γ∈C. These results are then extended for approximating roots of such polynomials for any given n. Such relations for the evaluation are motivated by the large computational efforts of the iterative numerical methods in solving for eigenvalues, and their errors. We first apply this relation to the eigenvalue problems represented by tridiagonal matrices with a periodicity k in its entries, providing a more accurate method for the evaluation of the spectra of a k-periodic chain and a reduction in computing effort from O(n2) to O(n). These results are combined with the spectral rules of Kronecker products, for efficient and accurate evaluation of spectra of spatial lattices and other periodic graphs that need not be represented by a banded or a Toeplitz type matrix.